High-order solutions around triangular libration points in the elliptic restricted three-body problem and applications to low energy transfers
High-order solutions around triangular libration points in the elliptic restricted three-body problem and applications to low energy transfers
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椭圆限制三体问题中三角振动点周围的高阶解及其在低能量传递中的应用
DOI：
10.1016/j.cnsns.2014.01.019
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发表时间：
2014-09
期刊：
影响因子：
--
通讯作者：
Bo Xu
中科院分区：
文献类型：
--
作者：
Hanlun Lei;Bo Xu
本文首先构造了椭圆约束三体问题(ERTBP)中三角点附近的高阶级数展开式，然后利用这些级数解设计了向地月系统三角点轨道的双冲低推力低能量传递。ERTBP在脉动汇合坐标系中的运动方程与圆形约束三体问题(CRTBP)中的运动方程具有相同的对称性，并具有5个平衡点。考虑到三角振动点的稳定动力学，ERTBP的运动解析解被表示为四个幅值的形式级数：初级、长、短和垂直周期幅值的轨道偏心率。利用Lindstedt-Poincaré方法构造了任意阶截断的级数展开式，从而可以将ERTBP中三角振动点周围的准周期轨道全部参数化。特别地，当基准点的偏心率为零时，所构造的级数展开式可以简化为描述CRTBP中三角振动点附近的运动。为了验证所构造的级数展开式的有效性，利用数值积分的方法研究了不同阶数对应的收敛域。在得到三角点周围有界轨道的解析解之后，实际飞行任务中的目标轨道可以用几个相关参数来表示。由于建造了一系列扩展，计划进行两次任务，将航天器从地球转移到地月系统三角平动点周围的短周期轨道。为了以更少的燃料成本完成任务，采用数值优化方法(包括全局优化技术和局部优化技术)对低能量传递(二冲量和低推力)进行了研究。模拟结果表明：(A)低推力、低能量传递在推进剂分数方面优于相应的二冲量、低能量传递；(B)与传统的类Hohmann传递相比，两冲量、低能量和低推力、低能量传递的效率都非常高，但以飞行时间为代价。
High-order series expansions around triangular libration points in the elliptic restricted three-body problem (ERTBP) are constructed first, and then with the aid of the series solutions, two-impulse and low-thrust low energy transfers to the triangular point orbits of the Earth–Moon system are designed in this paper. The equations of motion of ERTBP in the pulsating synodic reference frame have the same symmetries as the ones in circular restricted three-body problem (CRTBP), and also have five equilibrium points. Considering the stable dynamics of triangular libration points, the analytical solutions of the motion around them in ERTBP are expressed as formal series of four amplitudes: the orbital eccentricity of the primary, the long, short and vertical periodic amplitudes. The series expansions truncated at arbitrary order are constructed by means of Lindstedt–Poincaré method, and then the quasi-periodic orbits around triangular libration points in ERTBP can all be parameterized. In particular, when the eccentricity of the primary is zero, the series expansions constructed can be reduced to describe the motion around triangular libration points in CRTBP. In order to check the validity of the series expansions constructed, the domain of convergence corresponding to different orders is studied by using numerical integration. After obtaining the analytical solutions of the bounded orbits around triangular points, the target orbits in practical missions can be expressed by several related parameters. Thanks to the series expansions constructed, two missions are planned to transfer a spacecraft from the Earth to the short periodic orbits around triangular libration points of Earth–Moon system. To complete the missions with less fuel cost, low energy transfers (two-impulse and low-thrust) are investigated by means of numerical optimization methods (both global and local optimization techniques). Simulation results indicate that (a) the low-thrust, low energy transfers outperform the corresponding two-impulse, low energy transfers in terms of propellant fraction, and (b) compared with the traditional Hohmann-like transfers, both the two-impulse, low energy and low-thrust, low energy transfers perform very efficiently, at the cost of flight time.
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作者：
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